The nth row of Pascal’s triangle contains the binomial coefficients C(n, r) for r ranging from 0 to n. For large n, if you print out the numbers in the nth row vertically in binary you can see a circular arc.
Here’s an example with n = 50.
You don’t have to use binary. Here’s are the numbers in the row for n = 100 written in base 10. You may be read the numbers if you squint, but you can see that the shape of the curve is something like a piece of a circle.
The length of the numerical representation of a number is roughly proportional to its logarithm. Changing the base only changes the proportionality constant. The examples above suggests that a plot of the logarithms of a row of Pascal’s triangle will be a portion of a circle, up to some scaling of one of the axes, so in general we have an ellipse.
Here’s a plot of log C(n, r) for n = 1000. The shape is the same for all large n, so the choice of n = 1000 doesn’t matter much at all.
The best fitting ellipse is
where a = 554.2. b = 47.12, and y0 = −19.87.
The plots of log C(1000, r) and the ellipse lie on top of each other; the error is less than the width of a plotting line. Here’s a plot of the relative error in approximating log C(1000, r) with the ellipse.
Parabolic fit
WoЇfgang pointed out that the curve should be a parabola rather than an ellipse because the binomial distribution is asymptotically normal. Makes perfect sense.
So I redid my plots with the parabola that interpolates log C(n, r) at 0, n/2, and n. This also gives a very good fit, but not as good!
But that’s not a fair comparison because it’s comparing the best (least squares) elliptical fit to a convenient parabolic fit.
So I redid my plots again with the least squares parabolic fit. The fit was better, but still not as good as the elliptical fit.
I think the reason the ellipse fits better than the parabola has to do with the limitations of the central limit theorem. First of all, it applies to CDFs, not PDFs. Second, it applies to absolute error, not relative error. In practice, the CLT gives a good approximation in the middle but not in the tails. With all the curves mentioned above, the maximum error is in the tails.